State variable time response

Fig. 8.5 State variable time response and state trajectory for Example 8.4. Solution...

Fig. 10.15 Inverted pendulum state variable time response for three control strategies.

The time response of the state variables (i.e. position and veloeity) together with the state trajeetory is given in Figure 8.5. 

Figure 10.15 shows the time response of the inverted pendulum state variables from an initial eondition of 0 = 0.1 radians. On eaeh graph, three eontrol strategies are shown, the 11 set rulebase of equation (10.52), the 22 set rulebase that ineludes X and X, and the state feedbaek method given by equation (10.51). 

The for-end loop in examp88.m that employs equation (8.76), while appearing very simple, is in faet very powerful sinee it ean be used to simulate the time response of any size of multivariable system to any number and manner of inputs. If A and B are time-varying, then A(r) and B(r) should be ealeulated eaeh time around the loop. The author has used this teehnique to simulate the time response of a 14 state-variable, 6 input time-varying system. Example 8.10 shows the ease in whieh the eontrollability and observability matriees M and N ean be ealeulated using c t r b and ob s v and their rank eheeked. 

This paper extends previous studies on the control of a polystyrene reactor by including (1) a dynamic lag on the manipulated flow rate to improve dynamic decoupling, and (2) pole placement via state variable feedback to improve overall response time. Included from the previous work are optimal allocation of resources and steady state decoupling. Simulations on the non-linear reactor model show that response times can be reduced by a factor of 6 and that for step changes in desired values the dynamic decoupling is very satisfactory. 

The present paper applies state variable techniques of modern control theory to the process. The introduction of a dynamic transfer function to manipulate flow rate removes much of the transient fluctuations in the production rate. Furthermore, state variable feedback with pole placement improves the speed of response by about six times. 

The process response is presented in Figure 4.6. Observe that all the state variables exhibit a fast transient, followed by a slow approach to steady state, which is indicative of the two-time-scale behavior of the system, and is consistent with our observation that processes with impurities and purge are modeled by systems of ODEs that are in a nonstandard singularly perturbed form. 

The responses of all the state variables (Figure 5.7) exhibit an initial fast transient, followed by a slower dynamics. The states approach their nominal steady-state values after a period of time that exceeds 48 h (nota bene, two days ), indicating that a very slow component is also present in the process dynamics. The analysis in the following section will use the framework developed earlier in the chapter to provide a theoretical explanation for these findings. 

The most important variables in the system are the state variables, since it is their evolving behaviour in time that is the basis of the dynamic response of the system. The importance of their role may be brought out further by rearranging the equations in Section 2.1 to eliminate all the algebraic equations and leave just the four state equations, integration of which enables us to trace the response of the system. 

Physiological disturbances are such factors as consumption of metabolic substrates or tissue damage, while the psychological disturbance factors are those related to, for example, anxiety about work or inadequate social support. Changes in the state variables of the worker are defined by the model as responses. A response is an effect of the dose caused by exposure. For example, hand exertion can cause elastic deformation of tendons and changes in tissue composition and/or shape, which in turn may result in hand discomfort. The dose-response time relationship implies that the effect of a dose can be immediate or the response may be delayed for a long periods of time. 

FIGURE 22.3 Circuit representation of membrane current. The conductance can be nonlinear as indicated by the powers p and q on the state variables m and hi, respectively. These state variables are typically time-varying functions of the transmembrane potential difference, V. The battery, , represents the electrochemical gradient of the ionic species responsible for the current. 

The open-loop strategy implies that each players control is only a function of time, Ui = Ui t). A feedback strategy implies that each players control is also a function of state variables, ui = Ui t Xi t) Xj(t)). As in the static games, NE is obtained as a fixed point of the best response mapping by simultaneously solving a system of first-order optimality conditions for the players. Recall that to find the optimal control we first need to form a Hamiltonian. If we were to solve two individual non-competitive optimization problems, the Hamiltonians would be Hi = fi XiQi, i = 1,2, where Xi t) is an adjoint multiplier. However, with two players we also have to account for the state variable of the opponent so that the Hamiltonian becomes... 

In the present paper, the non-steady state simulation of a large-scale radial-flow ammonia converter is carried out, aiming to study the influence of the operating variables on the dynamic and stability of the reactor. By means of a detailed mathematical model, the time responses of the main variables are obtained and compared for two different conditions open loop (without feedback control) and closed loop. 

In summary, to calculate the activity of each element, first the time response of the state variables is determined by integrating the state equations in (2.1). Then,... 

Due to the simplicity of this model the state and output equations are derived by hand and transferred into MATLAB [21]. This step could have been done using a bond graph modeling environment (e.g., 20-Sim [22], CAMP-G [23]) to generate the time responses needed for the Analyze Activity step of MORA (see Fig. 2.2). The dynamic equations are numerically integrated to first produce the time response of the state variables and then the required set of outputs as defined in (2.3). 

Analytical techniques generally involve two areas. The first is the direct solution of the system differential equations in the Aime domain, usually by state variables. The second area is optimization of a specific performance criterion. The criteria for optimization by analytical techniques usually involve minimum response time or an integral time-cost function. 

In a static model, the most recent output or dependent variables depend on the most recent values of the input or independent variables. In dynamic models, the state variables describe the change in dependent variable as a function of independent variables and time, the response is ealled the system transient. In some eases, dynamic modeling may be required, for example when predicting future values of variables. Another example is the modeling of a batch reactor. In this type of reactor, the conditions change with time, hence a static model would not be very useful. When a continuous process is operated at constant conditions, however, the description of the independent variables eonld very well be static, in which case the time derivative does not appear in the model. 

Early studies of the failure process in glassy polymers, crazing, placed emphasis on the idea that one could separate stress effects frail strain effects by taking advantage of the face that even in the glassy state the mecheuiical response was strongly time dependent. Further studies in the area of variable rate impact... 

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